What you’re describing sounds like the “round ligament” pains of pregnancy. I’ve heard many of my patients describe these very pains.
When the uterus (womb) is the size of a pear in the non-pregnant state, the round ligaments help hold the uterus in place in the pelvis. But as the uterus stretches out during pregnancy, they never recover their supportive strength after delivery. During this stretching out process—and believe me the ligaments are very stretched out at 35 weeks (the pain can begin as early as 20 weeks)—one can experience this type of pain along the course of the ligaments.
The pair of round ligaments originate from the sides of the uterus, travel to the pelvic side walls where they enter the inquinal rings, and from there extend to the vaginal side walls and even the inner thighs. Laying on one side or the other will take tension off of the one that’s hurting as the uterus is allowed to “flop” the other way.
It is a harmless pain, but can be confused with contractions, so you should report this to your health care provider. Similar pains are experienced in the lower back—the “uterosacral” ligament pain—just as harmless. I’m afraid that round ligament pain is one of the phenomena of trying to cram two people into the space of one.